What is it about the elfin Mary-Kate Olsen that has caused the NYPD to misplace the common sense and good judgment that characteristically inform its investigations?

Is Police Commissioner Ray Kelly scared that she’s going to beat him up?

Or are the department’s lawyers afraid that Mary-Kate’s barristers are going to beat them up?

Thursday, the department was planning to ask the petite performer what she knows about the death of actor Heath Ledger – ask her directly, that is.

How do we know this?

For one thing, NYPD sources said so. (More on that below.)

Secondly, such interviews are fundamental police work – and nobody’s better at the fundamentals than New York’s Finest.

And while it appears that there is no great mystery remaining to Ledger’s death, one striking oddity is that one of the women who found his body called Olsen in California threetimes before calling 911, and then once right afterward.

And Olsen, rather than calling 911 herself, dispatched some New York-based bodyguards to Ledger’s SoHo apartment. To what purpose, we wonder.

So why wouldn’t the police want to interview Olsen?

Ah, now that is a mystery.

Again, we know that the cops intended to talk with Olsen – at least they did late Thursday – because officers working the case said so.

On Friday, however, came an abrupt reversal of course.

The department said that it had no plans to talk to Olsen – and that it never intended to do so.

To which we say: Bushwa!

Or, as Post Editor Col Allan more politely put it yesterday afternoon: “We confirmed this story [Thursday] night with an impeccable source inside the NYPD, and we stand by our reporting.”

But, again, why would the department so suddenly go so weak-kneed?

Here’s a possibility: From the outset, The Post has been receiving dark communications from Olsen’s lawyer, implying consequences if the young actress’ interests are bruised.

It’s no great leap to suspect that the NYPD has been targeted similarly – threatening people is what some lawyers do best.

If so, it’s a pity that the department caved.

That’s because, in so doing, Kelly & Co. have cast shadows of suspicion over what had seemed to be a sadly routine tragedy. There’s nothing new in the collision of an apparently troubled young celebrity with an abundance of prescription drugs.

On the other hand, it is decidedly not routine for the NYPD to leave loose ends hanging when it closes a case.

One Police Plaza needs to explain why it seemingly intends to do so in this one.